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polynomial sequence, writing numbers never stops

Source: IMOC 2020 A4

August 12, 2021
algebrapolynomial

Problem Statement

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers a−1,…,a−ka_{-1},\ldots,a_{-k} on a blackboard, so he decided to do the following process just for fun: if there are real numbers a−k,…,an−1a_{-k},\ldots,a_{n-1} on the blackboard, then he computes the polynomial Pn(t)=(1−a−kt)⋯(1−an−1t).P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t). He then writes a real number ana_n, where an=iPn(i)−iPn(−i)Pn(i)+Pn(−i).a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}. If ana_n is undefined (that is, Pn(i)+Pn(−i)=0P_n(i)+P_n(-i)=0), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists m>n≥0m>n\ge0 such that am=anam=an), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)
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